The quantitative behavior of asymptotic syzygies for Hirzebruch surfaces

Abstract

We study the quantitative behavior of asymptotic syzygies for certain toric surfaces, including Hirzebruch surfaces. In particular, we show that the asymptotic linear syzygies of Hirzebruch surfaces embedded by $𝒪(d,2)$ conform to Ein, Erman, and Lazarsfeld’s normality heuristic. We also show that the higher degree asymptotic syzygies are not asymptotically normally distributed.